Equilibrium Term Structure Models. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 854

Transcription

1 Equilibrium Term Structure Models c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 854

2 8. What s your problem? Any moron can understand bond pricing models. Top Ten Lies Finance Professors Tell Their Students c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 855

3 Introduction This chapter surveys equilibrium models. Since the spot rates satisfy r(t, T ) = ln P (t, T ), T t the discount function P (t, T ) suffices to establish the spot rate curve. All models to follow are short rate models. Unless stated otherwise, the processes are risk-neutral. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 856

4 The short rate follows The Vasicek Model a dr = β(µ r) dt + σ dw. The short rate is pulled to the long-term mean level µ at rate β. Superimposed on this pull is a normally distributed stochastic term σ dw. Since the process is an Ornstein-Uhlenbeck process, β(t t) E[ r(t ) r(t) = r ] = µ + (r µ) e from Eq. (53) on p a Vasicek (1977). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 857

5 The Vasicek Model (continued) The price of a zero-coupon bond paying one dollar at maturity can be shown to be P (t, T ) = A(t, T ) e B(t,T ) r(t), (97) where A(t, T ) = [ exp [ exp (B(t,T ) T +t)(β 2 µ σ 2 /2) β 2 σ2 B(t,T ) 2 4β σ 2 (T t) 3 6 ] ] if β 0, if β = 0. and B(t, T ) = β(t t) 1 e β if β 0, T t if β = 0. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 858

6 The Vasicek Model (concluded) If β = 0, then P goes to infinity as T. Sensibly, P goes to zero as T if β 0. Even if β 0, P may exceed one for a finite T. The spot rate volatility structure is the curve ( r(t, T )/ r) σ = σb(t, T )/(T t). When β > 0, the curve tends to decline with maturity. The speed of mean reversion, β, controls the shape of the curve. Indeed, higher β leads to greater attenuation of volatility with maturity. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 859

7 Yield 0.2 normal humped inverted Term c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 860

8 The Vasicek Model: Options on Zeros a Consider a European call with strike price X expiring at time T on a zero-coupon bond with par value $1 and maturing at time s > T. Its price is given by a Jamshidian (1989). P (t, s) N(x) XP (t, T ) N(x σ v ). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 861

9 The Vasicek Model: Options on Zeros (concluded) Above x 1 σ v ln ( P (t, s) ) P (t, T ) X + σ v 2, σ v v(t, T ) B(T, s), σ 2 [1 e 2β(T t) ] v(t, T ) 2 2β, if β 0 σ 2 (T t), if β = 0. By the put-call parity, the price of a European put is XP (t, T ) N( x + σ v ) P (t, s) N( x). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 862

10 Binomial Vasicek Consider a binomial model for the short rate in the time interval [ 0, T ] divided into n identical pieces. Let t T/n and p(r) β(µ r) t. 2σ The following binomial model converges to the Vasicek model, a r(k + 1) = r(k) + σ t ξ(k), 0 k < n. a Nelson and Ramaswamy (1990). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 863

11 Binomial Vasicek (continued) Above, ξ(k) = ±1 with Prob[ ξ(k) = 1 ] = p(r(k)) if 0 p(r(k)) 1 0 if p(r(k)) < 0 1 if 1 < p(r(k)). Observe that the probability of an up move, p, is a decreasing function of the interest rate r. This is consistent with mean reversion. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 864

12 Binomial Vasicek (concluded) The rate is the same whether it is the result of an up move followed by a down move or a down move followed by an up move. The binomial tree combines. The key feature of the model that makes it happen is its constant volatility, σ. For a general process Y with nonconstant volatility, the resulting binomial tree may not combine. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 865

13 The Cox-Ingersoll-Ross Model a It is the following square-root short rate model: dr = β(µ r) dt + σ r dw. (98) The diffusion differs from the Vasicek model by a multiplicative factor r. The parameter β determines the speed of adjustment. The short rate can reach zero only if 2βµ < σ 2. See text for the bond pricing formula. a Cox, Ingersoll, and Ross (1985). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 866

14 Binomial CIR We want to approximate the short rate process in the time interval [ 0, T ]. Divide it into n periods of duration t T/n. Assume µ, β 0. A direct discretization of the process is problematic because the resulting binomial tree will not combine. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 867

15 Binomial CIR (continued) Instead, consider the transformed process x(r) 2 r/σ. It follows dx = m(x) dt + dw, where m(x) 2βµ/(σ 2 x) (βx/2) 1/(2x). Since this new process has a constant volatility, its associated binomial tree combines. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 868

16 Binomial CIR (continued) Construct the combining tree for r as follows. First, construct a tree for x. Then transform each node of the tree into one for r via the inverse transformation r = f(x) x 2 σ 2 /4 (p. 870). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 869

17 x + 2 t f(x + 2 t) x + t f(x + t) x x f(x) f(x) x t f(x t) x 2 t f(x 2 t) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 870

18 Binomial CIR (concluded) The probability of an up move at each node r is p(r) β(µ r) t + r r r + r. (99) r + f(x + t) denotes the result of an up move from r. r f(x t) the result of a down move. Finally, set the probability p(r) to one as r goes to zero to make the probability stay between zero and one. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 871

19 Consider the process, Numerical Examples 0.2 (0.04 r) dt r dw, for the time interval [ 0, 1 ] given the initial rate r(0) = We shall use t = 0.2 (year) for the binomial approximation. See p. 873(a) for the resulting binomial short rate tree with the up-move probabilities in parentheses. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 872

20 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 873

21 Numerical Examples (continued) Consider the node which is the result of an up move from the root. Since the root has x = 2 r(0)/σ = 4, this particular node s x value equals 4 + t = Use the inverse transformation to obtain the short rate x 2 (0.1) 2 / c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 874

22 Numerical Examples (concluded) Once the short rates are in place, computing the probabilities is easy. Note that the up-move probability decreases as interest rates increase and decreases as interest rates decline. This phenomenon agrees with mean reversion. Convergence is quite good (see text). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 875

23 A General Method for Constructing Binomial Models a We are given a continuous-time process dy = α(y, t) dt + σ(y, t) dw. Make sure the binomial model s drift and diffusion converge to the above process by setting the probability of an up move to α(y, t) t + y y d y u y d. Here y u y + σ(y, t) t and y d y σ(y, t) t represent the two rates that follow the current rate y. The displacements are identical, at σ(y, t) t. a Nelson and Ramaswamy (1990). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 876

24 A General Method (continued) But the binomial tree may not combine: σ(y, t) t σ(y u, t) t σ(y, t) t + σ(y d, t) t in general. When σ(y, t) is a constant independent of y, equality holds and the tree combines. To achieve this, define the transformation x(y, t) y σ(z, t) 1 dz. Then x follows dx = m(y, t) dt + dw for some m(y, t) (see text). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 877

25 A General Method (continued) The key is that the diffusion term is now a constant, and the binomial tree for x combines. The probability of an up move remains α(y(x, t), t) t + y(x, t) y d (x, t), y u (x, t) y d (x, t) where y(x, t) is the inverse transformation of x(y, t) from x back to y. Note that y u (x, t) y(x + t, t + t) and y d (x, t) y(x t, t + t). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 878

26 A General Method (concluded) The transformation is r (σ z) 1 dz = 2 r/σ for the CIR model. The transformation is S (σz) 1 dz = (1/σ) ln S for the Black-Scholes model. The familiar binomial option pricing model in fact discretizes ln S not S. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 879

27 Model Calibration In the time-series approach, the time series of short rates is used to estimate the parameters of the process. This approach may help in validating the proposed interest rate process. But it alone cannot be used to estimate the risk premium parameter λ. The model prices based on the estimated parameters may also deviate a lot from those in the market. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 880

28 Model Calibration (concluded) The cross-sectional approach uses a cross section of bond prices observed at the same time. The parameters are to be such that the model prices closely match those in the market. After this procedure, the calibrated model can be used to price interest rate derivatives. Unlike the time-series approach, the cross-sectional approach is unable to separate out the interest rate risk premium from the model parameters. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 881

29 On One-Factor Short Rate Models By using only the short rate, they ignore other rates on the yield curve. Such models also restrict the volatility to be a function of interest rate levels only. The prices of all bonds move in the same direction at the same time (their magnitudes may differ). The returns on all bonds thus become highly correlated. In reality, there seems to be a certain amount of independence between short- and long-term rates. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 882

30 On One-Factor Short Rate Models (continued) One-factor models therefore cannot accommodate nondegenerate correlation structures across maturities. Derivatives whose values depend on the correlation structure will be mispriced. The calibrated models may not generate term structures as concave as the data suggest. The term structure empirically changes in slope and curvature as well as makes parallel moves. This is inconsistent with the restriction that all segments of the term structure be perfectly correlated. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 883

31 On One-Factor Short Rate Models (concluded) Multi-factor models lead to families of yield curves that can take a greater variety of shapes and can better represent reality. But they are much harder to think about and work with. They also take much more computer time the curse of dimensionality. These practical concerns limit the use of multifactor models to two-factor ones. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 884

32 Options on Coupon Bonds a The price of a European option on a coupon bond can be calculated from those on zero-coupon bonds. Consider a European call expiring at time T on a bond with par value $1. Let X denote the strike price. The bond has cash flows c 1, c 2,..., c n t 1, t 2,..., t n, where t i > T for all i. at times The payoff for the option is ( n ) max c i P (r(t ), T, t i ) X, 0. a Jamshidian (1989). i=1 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 885

33 Options on Coupon Bonds (continued) At time T, there is a unique value r for r(t ) that renders the coupon bond s price equal the strike price X. This r can be obtained by solving X = n i=1 c ip (r, T, t i ) numerically for r. The solution is also unique for one-factor models whose bond price is a monotonically decreasing function of r. Let X i P (r, T, t i ), the value at time T of a zero-coupon bond with par value $1 and maturing at time t i if r(t ) = r. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 886

34 Options on Coupon Bonds (concluded) Note that P (r(t ), T, t i ) X i if and only if r(t ) r. As X = i c ix i, the option s payoff equals n c i max(p (r(t ), T, t i ) X i, 0). i=1 Thus the call is a package of n options on the underlying zero-coupon bond. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 887

35 No-Arbitrage Term Structure Models c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 888

36 How much of the structure of our theories really tells us about things in nature, and how much do we contribute ourselves? Arthur Eddington ( ) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 889

37 Motivations Recall the difficulties facing equilibrium models mentioned earlier. They usually require the estimation of the market price of risk. They cannot fit the market term structure. But consistency with the market is often mandatory in practice. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 890

38 No-Arbitrage Models a No-arbitrage models utilize the full information of the term structure. They accept the observed term structure as consistent with an unobserved and unspecified equilibrium. From there, arbitrage-free movements of interest rates or bond prices over time are modeled. By definition, the market price of risk must be reflected in the current term structure; hence the resulting interest rate process is risk-neutral. a Ho and Lee (1986). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 891

39 No-Arbitrage Models (concluded) No-arbitrage models can specify the dynamics of zero-coupon bond prices, forward rates, or the short rate. Bond price and forward rate models are usually non-markovian (path dependent). In contrast, short rate models are generally constructed to be explicitly Markovian (path independent). Markovian models are easier to handle computationally. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 892

40 The Ho-Lee Model a The short rates at any given time are evenly spaced. Let p denote the risk-neutral probability that the short rate makes an up move. We shall adopt continuous compounding. a Ho and Lee (1986). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 893

41 r 2 + v 2 r 3 r 2 r 1 r 3 + v 3 r 3 + 2v 3 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 894

42 The Ho-Lee Model (continued) The Ho-Lee model starts with zero-coupon bond prices P (t, t + 1), P (t, t + 2),... at time t identified with the root of the tree. Let the discount factors in the next period be P d (t + 1, t + 2), P d (t + 1, t + 3),... P u (t + 1, t + 2), P u (t + 1, t + 3),... if short rate moves down if short rate moves up By backward induction, it is not hard to see that for n 2, (see text). P u (t + 1, t + n) = P d (t + 1, t + n) e (v 2+ +v n ) (100) c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 895

43 The Ho-Lee Model (continued) It is also not hard to check that the n-period zero-coupon bond has yields y d (n) ln P d(t + 1, t + n) n 1 y u (n) ln P u(t + 1, t + n) n 1 = y d (n) + v v n n 1 The volatility of the yield to maturity for this bond is therefore κ n py u (n) 2 + (1 p) y d (n) 2 [ py u (n) + (1 p) y d (n) ] 2 = p(1 p) (y u (n) y d (n)) = p(1 p) v v n n 1. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 896

44 The Ho-Lee Model (concluded) In particular, the short rate volatility is determined by taking n = 2: σ = p(1 p) v 2. (101) The variance of the short rate therefore equals p(1 p)(r u r d ) 2, where r u and r d are the two successor rates. a a Contrast this with the lognormal model. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 897

45 The Ho-Lee Model: Volatility Term Structure The volatility term structure is composed of κ 2, κ 3,.... It is independent of the r i. It is easy to compute the v i s from the volatility structure, and vice versa. The r i s can be computed by forward induction. The volatility structure is supplied by the market. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 898

46 The Ho-Lee Model: Bond Price Process In a risk-neutral economy, the initial discount factors satisfy P (t, t+n) = (pp u (t+1, t+n)+(1 p) P d (t+1, t+n)) P (t, t+1). Combine the above with Eq. (100) on p. 895 and assume p = 1/2 to obtain a P d (t + 1, t + n) = P u (t + 1, t + n) = P (t, t + n) P (t, t + 1) P (t, t + n) P (t, t + 1) 2 exp[ v v n ] 1 + exp[ v v n ], (102) exp[ v v n ]. (102 ) a In the limit, only the volatility matters. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 899

47 The Ho-Lee Model: Bond Price Process (concluded) The bond price tree combines. Suppose all v i equal some constant v and δ e v > 0. Then P d (t + 1, t + n) = P u (t + 1, t + n) = P (t, t + n) P (t, t + 1) P (t, t + n) P (t, t + 1) 2δ n δ n 1, δ n 1. Short rate volatility σ equals v/2 by Eq. (101) on p Price derivatives by taking expectations under the risk-neutral probability. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 900

48 The Ho-Lee Model: Yields and Their Covariances The one-period rate of return of an n-period zero-coupon bond is ( ) P (t + 1, t + n) r(t, t + n) ln. P (t, t + n) Its value is either ln P d(t+1,t+n) P (t,t+n) Thus the variance of return is or ln P u(t+1,t+n) P (t,t+n). Var[ r(t, t + n) ] = p(1 p)((n 1) v) 2 = (n 1) 2 σ 2. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 901

49 The Ho-Lee Model: Yields and Their Covariances (concluded) The covariance between r(t, t + n) and r(t, t + m) is (n 1)(m 1) σ 2 (see text). As a result, the correlation between any two one-period rates of return is unity. Strong correlation between rates is inherent in all one-factor Markovian models. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 902

50 The Ho-Lee Model: Short Rate Process The continuous-time limit of the Ho-Lee model is dr = θ(t) dt + σ dw. This is Vasicek s model with the mean-reverting drift replaced by a deterministic, time-dependent drift. A nonflat term structure of volatilities can be achieved if the short rate volatility is also made time varying, i.e., dr = θ(t) dt + σ(t) dw. This corresponds to the discrete-time model in which v i are not all identical. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 903

51 The Ho-Lee Model: Some Problems Future (nominal) interest rates may be negative. The short rate volatility is independent of the rate level. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 904

52 Problems with No-Arbitrage Models in General Interest rate movements should reflect shifts in the model s state variables (factors) not its parameters. Model parameters, such as the drift θ(t) in the continuous-time Ho-Lee model, should be stable over time. But in practice, no-arbitrage models capture yield curve shifts through the recalibration of parameters. A new model is thus born everyday. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 905

53 Problems with No-Arbitrage Models in General (concluded) This in effect says the model estimated at some time does not describe the term structure of interest rates and their volatilities at other times. Consequently, a model s intertemporal behavior is suspect, and using it for hedging and risk management may be unreliable. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 906

54 The Black-Derman-Toy Model a This model is extensively used by practitioners. The BDT short rate process is the lognormal binomial interest rate process described on pp. 747ff (repeated on next page). The volatility structure is given by the market. From it, the short rate volatilities (thus v i ) are determined together with r i. a Black, Derman, and Toy (BDT) (1990). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 907

55 r 4 r 3 r 2 r 4 v 4 r 1 r 3 v 3 r 2 v 2 r 4 v4 2 r 3 v 2 3 r 4 v 3 4 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 908

56 The Black-Derman-Toy Model (concluded) Our earlier binomial interest rate tree, in contrast, assumes v i are given a priori. A related model of Salomon Brothers takes v i to be constants. Lognormal models preclude negative short rates. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 909

57 The BDT Model: Volatility Structure The volatility structure defines the yield volatilities of zero-coupon bonds of various maturities. Let the yield volatility of the i-period zero-coupon bond be denoted by κ i. P u is the price of the i-period zero-coupon bond one period from now if the short rate makes an up move. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 910

58 The BDT Model: Volatility Structure (concluded) P d is the price of the i-period zero-coupon bond one period from now if the short rate makes a down move. Corresponding to these two prices are the following yields to maturity, y u P 1/(i 1) u 1, y d P 1/(i 1) d 1. The yield volatility is defined as κ i (1/2) ln(y u /y d ). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 911

59 The BDT Model: Calibration The inputs to the BDT model are riskless zero-coupon bond yields and their volatilities. For economy of expression, all numbers are period based. Suppose inductively that we have calculated r 1, v 1, r 2, v 2,..., r i 1, v i 1. They define the binomial tree up to period i 1. We now proceed to calculate r i and v i to extend the tree to period i. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 912

60 The BDT Model: Calibration (continued) Assume the price of the i-period zero can move to P u or P d one period from now. Let y denote the current i-period spot rate, which is known. In a risk-neutral economy, P u + P d 2(1 + r 1 ) = 1 (1 + y) i. (103) Obviously, P u and P d are functions of the unknown r i and v i. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 913

61 The BDT Model: Calibration (continued) Viewed from now, the future (i 1)-period spot rate at time one is uncertain. Let y u and y d represent the spot rates at the up node and the down node, respectively, with κ 2 denoting the variance, or ( ) κ i = 1 2 ln P 1/(i 1) u 1. (104) P 1/(i 1) d 1 c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 914

62 The BDT Model: Calibration (continued) We will employ forward induction to derive a quadratic-time calibration algorithm. a Recall that forward induction inductively figures out, by moving forward in time, how much $1 at a node contributes to the price (review p. 772(a)). This number is called the state price and is the price of the claim that pays $1 at that node and zero elsewhere. a Chen and Lyuu (1997); Lyuu (1999). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 915

63 The BDT Model: Calibration (continued) Let the unknown baseline rate for period i be r i = r. Let the unknown multiplicative ratio be v i = v. Let the state prices at time i 1 be P 1, P 2,..., P i, corresponding to rates r, rv,..., rv i 1, respectively. One dollar at time i has a present value of f(r, v) P r + P rv + P rv P i 1 + rv i 1. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 916

64 The BDT Model: Calibration (continued) The yield volatility is g(r, v) 1 2 ln ( Pu,1 1+rv + P u,2 1+rv ( Pd,1 1+r + P d,2 1+rv + + P u,i 1 1+rv i 1 ) 1/(i 1) 1 P d,i 1 1+rv i 2 ) 1/(i 1) 1. Above, P u,1, P u,2,... denote the state prices at time i 1 of the subtree rooted at the up node (like r 2 v 2 p. 908). on And P d,1, P d,2,... denote the state prices at time i 1 of the subtree rooted at the down node (like r 2 on p. 908). c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 917

65 The BDT Model: Calibration (concluded) Now solve f(r, v) = 1 (1 + y) i and g(r, v) = κ i for r = r i and v = v i. This O(n 2 )-time algorithm appears in the text. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 918

66 The BDT Model: Continuous-Time Limit The continuous-time limit of the BDT model is d ln r = (θ(t) + σ (t) σ(t) ln r) dt + σ(t) dw. The short rate volatility clearly should be a declining function of time for the model to display mean reversion. In particular, constant volatility will not attain mean reversion. c 2008 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 919